If the area of an equilateral triangle is `81sqrt(3) cm^(2)` , find its height .

Explanation

To find the height of an equilateral triangle given its area, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Area Formula for an Equilateral Triangle**:
The area \( A \) of an equilateral triangle with side length \( x \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} x^2 \]

2. **Set Up the Equation**:
We know the area of the triangle is \( 81\sqrt{3} \, \text{cm}^2 \). Thus, we can set up the equation:
\[ \frac{\sqrt{3}}{4} x^2 = 81\sqrt{3} \]

3. **Eliminate \(\sqrt{3}\)**:
To simplify the equation, we can divide both sides by \(\sqrt{3}\):
\[ \frac{1}{4} x^2 = 81 \]

4. **Multiply by 4**:
Next, we multiply both sides by 4 to isolate \( x^2 \):
\[ x^2 = 81 \times 4 \]

5. **Calculate \( x^2 \)**:
Now, calculate \( 81 \times 4 \):
\[ x^2 = 324 \]

6. **Find \( x \)**:
Take the square root of both sides to find \( x \):
\[ x = \sqrt{324} = 18 \, \text{cm} \]

7. **Use the Height Formula**:
The height \( h \) of an equilateral triangle can be found using the formula:
\[ h = \frac{\sqrt{3}}{2} x \]
Substitute \( x = 18 \, \text{cm} \):
\[ h = \frac{\sqrt{3}}{2} \times 18 \]

8. **Calculate the Height**:
Simplifying gives:
\[ h = 9\sqrt{3} \, \text{cm} \]

### Final Answer:
The height of the equilateral triangle is \( 9\sqrt{3} \, \text{cm} \).

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